In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous.
Definition
Let
be a compact subset of a metric space (such as
), and let
be a function from
into itself. The modulus of continuity of
is
![{\displaystyle \omega _{f}(t)=\sup _{d(x,y)\leq t}d(f(x),f(y)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9800cce80473301aa7503196d56be3e4324c8c)
The function
is called Dini-continuous if
![\int_0^1 \frac{\omega_f(t)}{t}\,dt < \infty.](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd7ef730b1454e242763d49e6f87fb94520d1ee)
An equivalent condition is that, for any
,
![\sum_{i=1}^\infty \omega_f(\theta^i a) < \infty](https://wikimedia.org/api/rest_v1/media/math/render/svg/50a9e2c8eb7582fb62800b8dc90952a3e4661789)
where
is the diameter of
.